UK Nonlinear News, July 1996

Recent Theses

Computer Aided Design and Analysis of Nonlinear Phenomena in Relay Control Systems

University of Sussex

Supervisor: Professor Derek P. Atherton (
Student: Dr Ali Moeini.


This thesis describes research on an extension of the Tsypkin method known as the A-function method for the determination of limit cycles in relay feedback systems. The A-function method has been studied thoroughly and several methods to calculate the A-function for any transfer function are given. The relationship between this method and the other methods such as the advanced z-transform method and the Hamel method are also given. Further extensions presented in this work show how limit cycles can be found in relay feedback loops with additional nonlinearities such as, a polynomial type and saturation, which operate on the feedback from the output of the plant and its successive derivatives. Some studies have been conducted on systems which are predicted to have multiple unstable limit cycles which is found to be a route to chaos and a conjecture for the occurrence of chaos is given. Moreover, the basin of attraction of some systems has been found through this method.

An exact orbital stability criterion for periodic motions in single loop relay systems has been formally presented and modified to examine more complex situations such as forced systems, asymmetric oscillations and mode dependent relay control systems.

The A-function method results in a set of nonlinear equations which are used to find the parameters of the limit cycle. Thus from a design point of view it has been necessary to implement the method as a package of computer programs which have been developed in conjunction with this thesis. The new software tool for the analysis and design of relay control systems has been implemented in Matlab which has a powerful graphic user interface. Graphical facilities of the program provide the A-locus of the plant and the negative reciprocal describing function of the relay from which the regions of frequency for the existence of limit cycles can easily be obtained. Analytical facilities of the program give the exact value of the frequencies of the limit cycles and the waveforms at the input of the relay and assess the stability of the limit cycles. The waveforms of the input to the relay are an important part of the solution procedure as they indicate whether it is valid or not. If the solution is invalid the waveforms often indicate the possibility of a more complex periodic mode. Moreover, various aspects of the dynamics of relay systems can be studied using this software such as chaotic motion, transient chaotic motion and the occurrence and analysis of multi-pulse oscillations. A general method has been obtained for the analysis of mode dependent relay systems, that is where the plant transfer function changes according to the output of the relay. This is a very practical problem for the control of many on/off type processes where the plant dynamics change according to whether the system is in the on or off mode. Many examples have been given and the results have been compared with the simulations and where appropriate, the describing function method.

Source: Professor Derek P Atherton (

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Last Updated: 23rd July 1996.